Hand calculating exp(x)
The previous post mentioned that Martin Gardner announced that Ramanujan's conjecture that exp(163) in an integer had been proven. This was an April Fool's joke in 1975. Gardner said
Working by hand, he [Ramanujan] found the value to be 262537412640768743.999999999999... The calculations were tedious, and he was unable to verify the next decimal digits.
Calculating exp(163) without a computer-Ramanujan died in 1920-would indeed be tedious, but not insurmountable. Certainly it would not stop someone like Ramanujan from testing a conjecture.
How might you go about calculating exp(163) by hand?
AlgorithmOne possibility is an algorithm in [1].
where r = t' / 256, t' = t - n log 2, and n is chosen to minimize |t'|.
We can choose n so that |t'| < log(2)/2 and so |r| < 0.014. This means the infinite series converges rapidly and not too many terms will be needed, depending on the desired precision.
The calculation x256 can be done by squaring 8 times.
Although the context of this post is hand calculations, this would also be a viable algorithm for a program doing extended precision calculations.
ExampleIn our case,t = 163 = 40.1091... and we choose n = 58 so that t' = -0.09336....
Then r = -0.000364....
So each term in the series will contribute 3 or 4 decimal places to the desired precision at first, more once the factorial denominators get large.
Related posts[1] Jonathan Borwein and David Bailey. Mathematics by Experiment. Volume 1.
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